<?xml version="1.0"?>
<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E31">Satz8p18Lemma</font></span>: <a NAME="T32"><span class="comment"><font color="firebrick">:: GTARSKI4:32</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="gtarski1.html#V9" title="GTARSKI1:attr.9">satisfying_Tarski-model</a>   <a href="gtarski1.html#L1" title="GTARSKI1:struct.1">TarskiGeometryStruct</a> <br/>  for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">c9</font>, <font color="Olive" title="b9">p</font>, <font color="Olive" title="b10">q</font>, <font color="Olive" title="b11">q9</font> being   <a href="gtarski1.html#NM1" title="GTARSKI1:NM.1">POINT</a> of <font color="Olive" title="b1">S</font>  st  not  <a href="gtarski1.html#R6" title="GTARSKI1:pred.6">Collinear</a> <font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b6">b</font>,<font color="Olive" title="b5">a</font>,<font color="Olive" title="b3">y</font> &amp; <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b5">a</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">z</font> &amp; <font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">z</font> <a href="gtarski1.html#R2" title="GTARSKI1:pred.2">equiv</a> <font color="Olive" title="b3">y</font>,<font color="Olive" title="b9">p</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b9">p</font> &amp; <font color="Olive" title="b11">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gtarski3.html#K3" title="GTARSKI3:func.3">reflection</a> (<font color="Olive" title="b4">z</font>,<font color="Olive" title="b10">q</font>) &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Olive" title="b7">c</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b8">c9</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b11">q9</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b8">c9</font> &amp;  <a href="gtarski3.html#R9" title="GTARSKI3:pred.9">Middle</a> <font color="Olive" title="b3">y</font>,<font color="Olive" title="b9">p</font>,<font color="Olive" title="b7">c</font> &amp;  <a href="gtarski1.html#R1" title="GTARSKI1:pred.1">between</a> <font color="Olive" title="b9">p</font>,<font color="Olive" title="b3">y</font>,<font color="Olive" title="b10">q</font> &amp; <font color="Olive" title="b10">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b11">q9</font> holds <br/><font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">y</font></div></div>
